RAW
No damage dice => No dice to double
Sucks, but them's the rules. So a critical hit with an unarmed strike does nothing more than 1 damage, excluding modifiers. (I'll be excluding those for the rest of the answer) The tables in the player's handbook doesn't list a "1d1", just 1 damage. "1" is not a dice roll, it's a number.
Of course, the designers do seem intent on making unarmed strikes do 1 damage. I doubt it's a typo, especially considering the basic rules have been updated (it's on v0.2). They even supplied a feat (tavern brawler in PHB) and a class (monk) which increases the unarmed strike damage from 1+mod to some actual dice (like a 1d4). If they never intended the damage of unarmed strikes to be so lame, why would they provide that feat and that class?
As a side note: a critical hit from any weapon causes two failed death saving throws... including unarmed strikes. This is possibly the only time they are on par with more traditional weapons.
Possible RAI and RAF; The Spirit of Critical Hits
Critical hits seem to be meant to do more damage than a normal hit. This is reflected in the doubling of an attacks damage dice. It makes you generally do more damage than normal, although there is a slim chance that you don't do as much on that strike. You can see the odds for some weapon dice values here.
If you're looking to extend this logic to critical hits, I suggest instantiating a house rule for upping it to a 1d4 or supply 2 damage on a critical hit. This could be a fun rule, and it may alleviate the disappointment your players feel while getting that critical hit while hitting something with their... fists? ("Unarmed Strike" is really vague!)
However, it does seem that the designers really did want unarmed strikes to not hit hard unless you've had some experience. Once again, this is because of the Tavern Brawler feat in PHB and the monk class. Is this because they want people to use weapons unless they're a monk or a tavern brawler? Were they afraid of things like the grapplemancer builds coming into 5e? Hard to say. You could ask wizards of the coast by emailing sageadvice@wizards.com.
A Note
5e greatly acknowledges the importance of house rules in D&D culture, so I would not stick with the RAW reading of this. In that article linked above, written by Jeremy Crawford, one of the two lead designers for 5e, says:
We expect DMs to depart from the rules when running a particular campaign or when seeking the greatest happiness for a certain group of players.
The RAW just is not fun and takes the joy out of a critical hit. Come up with a rule that seems right (doubling to 2 damage, or going to 1d4 damage, or something else) for your group. In the words of Jeremy Crawford:
Fifth edition now belongs to the thousands of groups playing it.
Modify 5e to have fun in your group!
You are overcomplicating things here.
As a monk, your unarmed strike damage is replaced with a d4 (second Martial Arts bullet point, PH78), and "[y]ou can use dexterity instead of strength for the attack and damage rolls of your unarmed strikes and monk weapons" (first bullet point), so you ignore the normal 1 + Str mod.
Also, monks are not using the two-weapon fighting rules, they're using the Martial Arts rules. Last bullet point says you can make an unarmed strike as a bonus action—which, according to the first two bullet points, is Dex + Prof bonus attack bonus, with a d4 + Dex damage roll (at 1st level).
When using the Martial Arts rules, you follow the rules for Martial Arts, not any other rules. The specific MA rules override the general rules you are citing from elsewhere.
Best Answer
You only apply the strength modifier once because unarmed strikes don't involve a weapon
Unarmed strikes are considered melee weapon attacks, but don't involve a weapon
Unarmed strikes are melee weapon attacks despite not being made with a weapon.
The errata for the PHB states:
This is even more clearly stated in the Sage Advice Compendium:
and
Unarmed strikes only follow the rule for unarmed strikes
There are rules for how to do an attack with a weapon and rules for attacks from spells, but no rules for a melee weapon attack that has neither a weapon or from a spell. Unarmed strikes, being weird, fall into a crack in the rules.
Specifically, the second rule you mentioned does not apply because it is only for attacks with a weapon.
This is the reason that unarmed strikes get their own rule:
So you use this rule (and only this rule) to calculate the damage for an unarmed strike.